Is a gradient system considered an ODE or PDE?

Question

When you have a system of the type

$$\dfrac{dx(t)}{dt} = \nabla V(x)$$

Is this considered an ODE or a PDE? Because you have a single derivative with respect to $t$ on the lefthand side, whereas on the right hand side you have $\begin{bmatrix} \dfrac{\partial V(x)}{\partial x_1} \\ \dfrac{\partial V(x)}{\partial x_2} \\\vdots \end{bmatrix}$

Answer 1

It is definitely an ODE. The reason is that the function $V$ in a gradient system is known, so it's derivative can be evaluated explicitly, so you get a system $\frac{dx}{dt}=f(x)$, where $f(x)=\nabla V(x)$. There do exist PDE's expressed in terms of gradients (like Maxwell's equations), but they are not referred to as gradient systems.